In this final section, we study the scattering sequences which contribute to the Gieseker index
\(\Omega _\infty (\gamma )\)  for two simple choices of charges, as function of the phase \(\psi \) .
The initial rays in these sequences turn out to jump at a subset of the critical
values (REF ), while the index remains unaffected. This is reminiscent of the `fake walls' encountered in earlier studies [1]}, [2]}, [3]},
where the structure of the bound state changes while the index remains unchanged.
